It is desirable to automate many industrial or other processes for a number of reasons. For example, the automatic control of the piloting of an airplane is desirable to avoid human fatigue and error over long flights. In many industrial processes, such as the operation of a factory, automatic control is preferable to human control since the number of variables and speed of reaction makes human control impractical. In other processes, such as the operation of a household furnace, the use of an automatic control is the only practical solution, since direct human control would be uneconomic.
In all of these control applications, the automatic controller must react to changes in the operating parameters which in turn will require modification of the input parameters. To take the example of a household thermostat, the decrease in temperature of the house (the "process value", y(t)) below a pre-set temperature (the "set point", y.sub.SP) causes the thermostat to activate the furnace (the "control action" u(t)). The activation of the furnace eventually causes the temperature in the house to rise beyond the pre-set temperature, causing the thermostat to de-activate the furnace.
In more complicated systems, a more complex response of the controller may be required. For example, where the initial error in the system is great or has been present for a long time, the response from the controller may be more drastic than where the initial error is slight or recent in time. Similarly where the error is changing rapidly with time, the response of the controller may be more drastic than if the error is changing slowly. Controllers which are able to react in this way are referred to as Proportional-Integral-Derivative, or PID controllers, and such controllers have dominated industrial controllers to date. These controllers work by examining the instantaneous error between the process value and the set point. The Proportional term causes a larger control action to be taken for a larger error. The Integral term causes a larger control action to be taken if the error has persisted for some time. The Derivative term supplements the control action if the error is changing rapidly with time.
The value of the P-I-D terms depend on characteristics of the process and must be tuned accordingly to yield satisfactory control. Properly tuned PID controllers provide adequate control for a large portion of industrial applications. The basic problem with PID control schemes is that the control action is based on instantaneous error between the process variables and the set point without anticipation of the long term effects of the present control action or of the effects of previous control actions to which the process has not yet responded. For example, a past control action may cause a bump in the process variable which in turn causes a new control action, causing future bumps. This results in cycling due to dead time.
There are many processes with time-variant or non-linear characteristics which are difficult to control with fixed parameter PID controllers. For example, pH control, vibratory feed control and level control in irregular vessels are non-linear, and variable gain systems, mixed reagent control and heat exchanger control are time-variable. Processes which have significant dead-time, such as effluent treatment, furnace temperature control, systems with long pipelines and compressible fluids control also cause problems for PID controllers. To handle these processes, which are often the most critical to the operation of a plant, controllers have evolved to include adaptive features.
Adaptive controllers were first developed in the early 1950's for the avionics industry when difficulties were encountered applying PID controllers to the task of auto-piloting aircraft, where it was necessary for the controller to adapt to changes in flight characteristics such as air speed, altitude and aircraft load.
Two basic types of control schemes can be characterized as adaptive. The first are compensator-based controllers. These are based on the automatic tuning of a regulator which compensates for the gain and frequency response characteristics of the process. Self-tuning and gain scheduling PID controllers are examples of compensator-based designs. These designs regulate an error, typically set point minus process value, to generate a control input to the plant which compensates for the plant dynamics so that the process remains stable and can respond to load and set point changes quickly and with minimal overshoot. Gain scheduling involves experimentation to determine optimal PID parameter values under a variety of operating conditions. The PID parameter values are stored and later recalled for use in the controller according to prevailing ambient factors by way of a custom scheduling scheme. Self-tuning PID controllers use a variety of techniques to automatically determine the optimal values for the PID tuning parameters such as the Ultimate Sensitivity method developed by Ziegler and Nichols, the reaction curve method, or model fitting algorithms. These techniques typically require the introduction of a disturbance and subsequent analysis of the process reaction. Often the disturbance cannot be tolerated while in production or difficulty arises trying to identify the process response due to other load disturbances. Where significant dead-times are present, PID controllers are typically tuned with smaller proportional gain and longer integral action than actually required in order to prevent process cycling at a cost of poor response to load disturbances and setpoint changes.
Process model based adaptive controllers (also called "true" or "polynomial" adaptive controllers) are based on the automatic adjustment of a mathematical model of the process which is used to calculate the actual control action required to obtain a desired process response. Such adaptive controllers are better suited for complex processes with significant dead-times, since the mathematical model can take account of dead-time and consider the effects of past control actions which have not yet appeared in the measurable process variable, and the long term consequences of the control action to be taken. Hereafter, compensator-based controllers will be referred to as self-tuning PID controllers and process model based adaptive controllers will be referred to simply as "adaptive controllers".
Existing adaptive controllers follow the following basic steps:
1. The process model is adjusted to correct for model errors and to respond to changes in process characteristics by fitting observed process responses to the process model.
2. The previous control action and observed process response are incorporated into the model.
3. The plant model is used to predict process response and calculate the required controller output to bring the process variable to the desired set point.
The degree to which the model is able to accurately represent the process determines the efficiency of the control performance. Currently some models are prepared experimentally by compiling details of the plant operation. Such models are extremely plant specific, so the controller based on them is not transferable to different plants. Other adaptive controllers are based on general process models which do not adequately represent the process characteristics. They are difficult to apply and of inconsistent performance.
Problems remain with existing adaptive controllers. Those which function adequately typically require for an adequate process model a detailed knowledge of the plant transfer function, (plant order, time constants, dead time) which is generally obtained experimentally. This adds to the expense of installation and reduces the ability to transfer the controller to other processes. Those controllers of general application do not have process models which accurately represent the process characteristics. There is a also a need for improved adaptive control of industrial processes due to economic pressure and environmental concerns which call for more efficient process control. There is therefore a need for improved adaptive control of industrial processes by adaptive controllers which do not require detailed process analysis of the process to be controlled prior to implementation and which can be used to control a wide variety of processes.
Recently advances have been made in the use of orthonormal functions to model and control plants. C. C. Zervos and G. A. Dumont have outlined in "Deterministic adaptive control based on Laguerre series representation", International Journal of Control, 1988, vol. 48, no. 6, p. 2333 (hereinafter "Zervos"), the modelling of the plant by an orthonormal Laguerre network put in state-space form. Although other orthonormal functions such as Legendre functions may be used, Laguerre functions present certain advantages for process control. This article postulates the following predictive control law for calculating the process value y(t) after d future sample points: EQU y(t+d)=y(t)+K.sup.T l(t)+.beta.u(t)
where l(t) is the state-space vector such that y(t)=C.sup.T l(t), u(t) is the system input, T is the sampling period, A is a lower triangular N.times.N matrix as follows: ##EQU1##
Improvements have also recently been made in the use of recursive least squares algorithms for recursive parameter estimation. M. E. Salgado, G. C. Goodwin and R. H. Middleton in "Exponential Forgetting and Resetting", International Journal of Control, 1988, Vol. 47, No. 2 p. 477 (hereinafter "Salgado") describe an exponential forgetting and resetting algorithm ("EFRA") suitable for tracking time-varying parameters which performs better than the standard recursive least square algorithm.